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Implications of verify(dim)#

This demo demonstrates how setting dim in PerfectModelEnsemble.verify() and HindcastEnsemble.verify() alters the research question and prediction skill obtained, especially for spatial fields.

What’s the purpose of dim?

dim is designed the same way as in xarray.Dataset.mean(). A given metric is calculated over dimension dim, i.e. the result does not contain dim anymore. dim can be a string or list of strings.

3 ways to calculate aggregated prediction skill

  • Spatially aggregate first, then verify

  • verify on each grid point, then aggregate spatially

  • verify over spatial, member and initialization directly

# linting
%load_ext nb_black
%load_ext lab_black
%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np
import xarray as xr

import climpred

/home/docs/checkouts/ TqdmWarning: IProgress not found. Please update jupyter and ipywidgets. See
  from .autonotebook import tqdm as notebook_tqdm
<climpred.options.set_options at 0x7effa4b92b80>

sample spatial initialized ensemble data and nomenclature#

We also have some sample output that contains gridded data on MPIOM grid:

  • initialized3d: The initialized ensemble dataset with

    • members (1, … , 4) denoted by m

    • inits (initialization years: 3014, 3061, 3175, 3237) denoted by i

    • lead years (1, …, 5) denoted by l.

    • x and j (think as longitude and latitude): spatial dimensions denotes by s.

  • control3d: The control dataset spanning over time (3000, …, 3049) [not used here].

In PerfectModelEnsemble.verify(), we calculate a metric on data(s,i,m,l) containing spatial dimensions s and ensemble dimension i,m,l over dimensions dim denoted by metric(data(s,i,m,l), dim).

# Sea surface temperature
initialized3d = climpred.tutorial.load_dataset("MPI-PM-DP-3D")
control3d = climpred.tutorial.load_dataset("MPI-control-3D")
v = "tos"
initialized3d["lead"].attrs = {"units": "years"}
/home/docs/checkouts/ RuntimeWarning: Engine 'cfgrib' loading failed:
Cannot find the ecCodes library
  warnings.warn(f"Engine {name!r} loading failed:\n{ex}", RuntimeWarning)
# Create climpred PerfectModelEnsemble object.
pm = (
    # .generate_uninitialized() # needed for uninitialized in reference
/home/docs/checkouts/ UserWarning: Assuming annual resolution starting Jan 1st due to numeric inits. Please change ``init`` to a datetime if it is another resolution. We recommend using xr.CFTimeIndex as ``init``, see
/home/docs/checkouts/ UserWarning: Assuming annual resolution starting Jan 1st due to numeric inits. Please change ``init`` to a datetime if it is another resolution. We recommend using xr.CFTimeIndex as ``init``, see

subselecting North Atlantic#

pm = pm.isel(x=slice(80, 150), y=slice(40, 70))
pm.get_initialized().mean(["lead", "member", "init"])[v].plot(
    yincrease=False, robust=True
<matplotlib.collections.QuadMesh at 0x7eff8f179a30>
verify_kwargs = dict(
    metric="pearson_r",  # 'rmse'
    reference=["persistence", "climatology"],  # "uninitialized",
dim = ["init", "member"]
spatial_dims = ["x", "y"]

Here, we use the anomaly correlation coefficient climpred.metrics._pearson_r(). Also try root-mean-squared-error climpred.metrics._rmse().

Spatial aggregate first, then verify#

Here, we first average the raw ensemble data over the spatial_dims, i.e. creating North Atlantic averaged SST timeseries. On this timeseries, we calculate the prediction skill with PerfectModelEnsemble.verify().

skill(l) = metric(\overline{data(s,i,m,l)}^s, i, m)

where \overline{data}^{dim} represents the average of data over dimension dim

This approach answers the question: “What’s the prediction skill of SSTs, which were averaged before over the North Atlantic? What is the skill at predicting average [metric] in [domain]?”

Used in Séférian et al. [2018], Spring and Ilyina [2020], Pohlmann et al. [2004].

Recommended by Goddard et al. [2013] section 2.1.4:

“We advocate verifying on at least two spatial scales: (1) the observational grid scales to which the model data is interpolated, and (2) smoothed or regional scales. The latter can be accomplished by suitable spatial-smoothing algorithms, such as simple averages or spectral filters.”

pm_aggregated = pm.mean(spatial_dims)
# see the NA SST timeseries
TypeError                                 Traceback (most recent call last)
Cell In[7], line 3
      1 pm_aggregated = pm.mean(spatial_dims)
      2 # see the NA SST timeseries
----> 3 pm_aggregated.plot(show_members=True)

File ~/checkouts/, in PredictionEnsemble.plot(self, variable, ax, show_members, cmap, x)
    484 if cmap is None:
    485     cmap = "tab10"
--> 486 return plot_ensemble_perfect_model(
    487     self, variable=variable, ax=ax, show_members=show_members, cmap=cmap
    488 )

File ~/checkouts/, in plot_ensemble_perfect_model(pm, variable, ax, show_members, cmap, x)
    351 _cmap =, initialized.init.size)
    353 for ii, i in enumerate(initialized.init.values):
--> 354     dsi = initialized.sel(init=i)
    355     if uninitialized_present:
    356         dsu = uninitialized.sel(init=i)

File ~/checkouts/, in DataArray.sel(self, indexers, method, tolerance, drop, **indexers_kwargs)
   1417 def sel(
   1418     self: T_DataArray,
   1419     indexers: Mapping[Any, Any] | None = None,
   1423     **indexers_kwargs: Any,
   1424 ) -> T_DataArray:
   1425     """Return a new DataArray whose data is given by selecting index
   1426     labels along the specified dimension(s).
   1525     Dimensions without coordinates: points
   1526     """
-> 1527     ds = self._to_temp_dataset().sel(
   1528         indexers=indexers,
   1529         drop=drop,
   1530         method=method,
   1531         tolerance=tolerance,
   1532         **indexers_kwargs,
   1533     )
   1534     return self._from_temp_dataset(ds)

File ~/checkouts/, in Dataset.sel(self, indexers, method, tolerance, drop, **indexers_kwargs)
   2504 """Returns a new dataset with each array indexed by tick labels
   2505 along the specified dimension(s).
   2562 DataArray.sel
   2563 """
   2564 indexers = either_dict_or_kwargs(indexers, indexers_kwargs, "sel")
-> 2565 query_results = map_index_queries(
   2566     self, indexers=indexers, method=method, tolerance=tolerance
   2567 )
   2569 if drop:
   2570     no_scalar_variables = {}

File ~/checkouts/, in map_index_queries(obj, indexers, method, tolerance, **indexers_kwargs)
    181         results.append(IndexSelResult(labels))
    182     else:
--> 183         results.append(index.sel(labels, **options))
    185 merged = merge_sel_results(results)
    187 # drop dimension coordinates found in dimension indexers
    188 # (also drop multi-index if any)
    189 # (.sel() already ensures alignment)

File ~/checkouts/, in PandasIndex.sel(self, labels, method, tolerance)
    471 else:
    472     try:
--> 473         indexer = self.index.get_loc(label_value)
    474     except KeyError as e:
    475         raise KeyError(
    476             f"not all values found in index {coord_name!r}. "
    477             "Try setting the `method` keyword argument (example: method='nearest')."
    478         ) from e

File ~/checkouts/, in CFTimeIndex.get_loc(self, key, method, tolerance)
    466     return self._get_string_slice(key)
    467 else:
--> 468     return pd.Index.get_loc(self, key, method=method, tolerance=tolerance)

TypeError: get_loc() got an unexpected keyword argument 'method'
pm_aggregated_skill = pm_aggregated.verify(dim=dim, **verify_kwargs)
/Users/aaron.spring/Coding/climpred/climpred/ UserWarning: Calculate persistence from lead=1 instead of lead=0 (recommended).
[<matplotlib.lines.Line2D at 0x14d4628f0>,
 <matplotlib.lines.Line2D at 0x14d4631c0>,
 <matplotlib.lines.Line2D at 0x14b69cf10>]

verify on each grid point, then aggregate#

Here, we first calculate prediction skill with PerfectModelEnsemble.verify() and then average the SST skill over the North Atlantic region.

skill(l) = \overline{metric(data(s,i,m,l), i, m)}^s

This approach answers the question: “What’s the prediction skill of North Altantic SST grid points averaged afterwards? What is the average skill of predicting [metric] in [domain]?”

Used in Frölicher et al. [2020].

grid_point_skill = pm.verify(dim=dim, **verify_kwargs)
grid_point_skill[v].plot(col="lead", row="skill", robust=True, yincrease=False)
/Users/aaron.spring/Coding/climpred/climpred/ UserWarning: Calculate persistence from lead=1 instead of lead=0 (recommended).
<xarray.plot.facetgrid.FacetGrid at 0x14d8f1180>
grid_point_skill_aggregated = grid_point_skill.mean(spatial_dims)

verify over each grid point, member and initialization directly#

Here, we directly calculate prediction skill with PerfectModelEnsemble.verify() over spatial_dims, member and init.

skill(l) = metric(data(s,i,m,l), s, i, m)

This approach answers the question: What’s the prediction skill of the North Atlantic SST pattern (calculated over all members and inits)? What is the spatial skill of predicting [metric] over [domain]?

Used in Becker et al. [2014], Fransner et al. [2020].

skill_all_at_once = pm.verify(dim=dim + spatial_dims, **verify_kwargs)
/Users/aaron.spring/Coding/climpred/climpred/ UserWarning: Calculate persistence from lead=1 instead of lead=0 (recommended).

This approach yields very similar results to first calculating prediction skill over the spatial_dims, i.e. calculating a pattern correlation and then doing a mean over ensemble dimensions member and init.

skill(l) = \overline{metric(data(s,i,m,l), s)}^{i,m}

skill_spatial_then_ensemble_mean = pm.verify(dim=spatial_dims, **verify_kwargs).mean(
/Users/aaron.spring/Coding/climpred/climpred/ UserWarning: Calculate persistence from lead=1 instead of lead=0 (recommended).


The three approaches yield very different results based on the dim keyword. Make you choose dim according to the question you want to answer. And always compare your initialized skill to the reference skills, which also heavily depend on dim. The small details matter a lot!

skills = xr.concat(
    [pm_aggregated_skill, grid_point_skill_aggregated, skill_all_at_once], "method"
).assign_coords(method=["skill of aggregated", "grid skill aggregated", "all directly"])
skills[v].plot(hue="method", col="skill")
<xarray.plot.facetgrid.FacetGrid at 0x14d7f4460>
skills[v].plot(col="method", hue="skill")
<xarray.plot.facetgrid.FacetGrid at 0x14df5b970>



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Filippa Fransner, François Counillon, Ingo Bethke, Jerry Tjiputra, Annette Samuelsen, Aleksi Nummelin, and Are Olsen. Ocean Biogeochemical Predictions—Initialization and Limits of Predictability. Frontiers in Marine Science, 2020. doi:10/gg22rr.


Thomas L. Frölicher, Luca Ramseyer, Christoph C. Raible, Keith B. Rodgers, and John Dunne. Potential predictability of marine ecosystem drivers. Biogeosciences, 17(7):2061–2083, April 2020. doi:10/ggr9mb.


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